3G2 MAJOR P. A. MACiMAHON ON THE THEORY OP PARTITIONS OF NUMBERS. 
If the parts be limited to i in number and to i in magnitude, we find 
n ^ ~ 
> 1 — a^x 
+. 1 
-(A^r 
1 Y'"' 
' 1 - . 1 - . 1 - ... 1 - 
1 
cu 
1- X 
1 — X . \ — ox . 1 — of ... 1 — S'/ 
a, 
»-i 
the well-known result. 
Art. 74. It is to be remarked that the generating function in question may also be 
written 
1 
1 - g 
n- 
~ 1 — a^rp:. 1 — ~ o: . \ or ... 1 - x 
c.. 
i-l 
in which we have to seek the coefficient of g\ 
This function reduces to 
i - g .1 - gx .1 - gor .1 - yor .. .1 - pf 
the well-known form. 
In general, when a generating function reduces to the product of factt^rs 
1 
1 - .r* ’ 
the part-magnitude being unrestricted, we obtain a product of factors 
1 
I — //■«* 
for the restricted case, and this is frequently exhibitable, as regards the coefficients 
of as a })roduct of factors 
\—of 
The fl function is not altered by the interchange of the letters i,j. 
Art. 75. If the successive parts of the partition are limited in magnitude by 
J [1 Jr • • • Jii 
numbers necessarily in descending order, the generating function is. 
n -1 - 1 
“ 1 - ap 
, , c., 
L — I —^ 
-A-''-’ 
'i-l 
a., 
1 -X 
1 -. 7 ; 
